In
this page we illustrate how MTF is used to characterize the
performance of film and lenses.

Green is for
geeks. Do you get excited by a good
equation? Were you passionate
about your college math classes? Then you're probably a math geek-- a
member
of a maligned and misunderstood but highly elite fellowship. The text
in
green is for you. If you're normal or mathematically challenged, you
may
skip these sections. You'll never know what you missed.

Film

Digital photographers may skim this section
quickly. Many of the films have been discontinued.

The
curve on the left is the MTF for Kodak
E100VS reversal (slide) film (similar to Elite Chrome 100). A
different
MTF is shown for each color layer: red, green, and blue (R, G, and B).
Green is the most important because the eye is far more
sensitive to it than to other colors. The green curve is close
to the MTF curve for Fujichrome
Provia 100F.

Most film MTF curves can be closely approximated by a function
known
as the Lorentzian,

MTFfilm(
f ) = 1/(1+( f/f50)2)

f is frequency; f50
is the frequency where MTF
= 0.5 = 50%. For the E100VS green layer, f50
=
40 line pairs/mm. For the mathematically challenged, MTF( f
)
is expressed
in functional notation, which indicates that MTF is
a function of
frequency, f.

Electrical
engineers will note that this
curve has a second order Butterworth response. Second order means that
at high frequencies, f
>> f50,
response
drops twice as fast as frequency increases; it is proportional to 1/ f
2.
This is also called this a 12 dB (decibels) per octave rolloff (6 dB
per
octave per order).

The MTFcurve program illustrates the effects of MTF on the input
target.
MTF is applied only in the horizontal direction (it's two dimensional
in
actual images).

The blue
curve below the target
is the film MTF, expressed in percentage-- the scale on the left
applies.
It closely approximates the green curve for Ektachrome E100VS. The
red
curve shows the density of the bar pattern. The 50% and 10% amplitudes
are 40 line pairs/mm, as expected, and 120 line pairs/mm.

This plot is
generated from Matlab by enteringMTFcurve2 40

.

The MTF curve on the left
is from an old data sheet for Fujichrome Velvia,
a highly saturated ASA 50 slide film known for its sharpness. Something
is different here: a boost in the MTF response, which peaks at around
125%
at 10 line pairs/mm. What can this mean?

From my former career, I recognized this curve as
characteristic
of pulse
slimming equalizers, widely used in older magnetic disk drives to
sharpen
readback pulses so more data can be squeezed into a given area. By
adding one more parameter, fboost, I was able to fit it reasonably
well.

MTFfilm(
f )
= k / (1+(( f - fboost)
/ f50a )2)

fboost
is the frequency of maximum MTF. The percentage boost depends on
the ratio, fboost /
f50. fboost
must
be less than f50
/2.
k is a constant set so that MTF(0) = 1. One caution
about this
equation:
it rolls off more rapidly at high spatial frequencies ( f
>> f50)
than the equation without fboost (or fboost = 0). This boost is called
the adjacency
effect boost for
reasons described below. It is similar in nature to the boost caused by
digital sharpening.

f50a =
sqrt( f502-2f50 fboost
- fboost2)
so that f50 (as
entered in MTFcurve) is still the
50% MTF frequency.

This plot on the right is generated
from Matlab
by entering,MTFcurve2 45 13

The striking feature of the response, shown in the red
curve below the image, is the exaggerated contrast boundaries. Tones
overshoot
their low frequency (steady state) levels, and the result is plainly
visible.
The overall effect is that this image appears slightly
sharper
than the image above for Ektachrome E100VS or Fuji Provia 100F, even
though
it has slightly lower response at 100 line pairs/mm.

This effect is similar to the sharpen or unsharp
mask functions in image
editing programs.

There is a fairly elementary explanation of how this
happens in film.
It it particularly noticeable with black & white films
developed in
highly diluted (one-shot) developers. Visualize a contrast boundary
between
two areas, one side of which is heavily exposed (a highlight) and the
other
lightly exposed (a shadow). As the film is developed (particularly in
the
interval between between agitations), the developer in the heavily
exposed
area becomes depleted more rapidly than the developer in the lightly
exposed
area, slowing down the development. Some of the developer diffuses
across
the boundary, so that the developer in the heavily exposed area
adjacent
to the boundary is less depleted than in the rest of the region, hence
develops more rapidly, and the developer in the lightly exposed area
adjacent
to the boundary is more depleted, hence develops more slowly. The
result
is an exaggerated contrast boundary, as illustrated above. This is
known
as theadjacency effect, and
film/developer
combinations that exhibit
it are said to have high acutance.
In Velvia, the diffusion probably takes place inside the film.

The adjacency effect increases the perceived image sharpness,
though it can become irritatingly obvious if it's overdone. It boosts
the
the MTF 50% frequency, but it has relatively little effect on the 10%
level,
which is related to perceived line pairs per mm resolution.
Developing
technique affects MTF, particularly for B&W films, where there
is a
large choice of developers and agitation procedures. Development
options
are more limited for color films.

..An
excellent opportunity to
collect high quality photographic prints and support this website

.

..

Is
color negative film sharper?

If you choose to believe manufacturer's data sheets, yes, they are sharper. Thanks
to Mitch Mirkin and Jussi Ikäheimo for
pointing out data
I'd missed. MTF data for Fujicolor
Superia 100 color negative film is shown on the right. f50
is around 63 lp/mm.
That's about 50%
better than Fuji's highly regarded Provia
100F slide film. (Curiously, resolving
power, which has less
to do with perceived sharpness, is about the same.) Kodak
color negative films show a comparable improvement, for example, about
72 cycles/mm for Ultra Color 100UC vs. about 38
cycles/mm for Kodachrome 64 and 40 cycles/mm
for Elite Chrome 100 and Chrome Extra Color 100. MTF for Kodak
Gold films (Farbwelt in German) are shown on the Kodak
Germany site. For
Gold 100,
shown below, f50
(extrapolated)
is about 110 lines/mm with a 40% adjacency effect boost. (Thanks to
Stefan
Ittner for directing me to the Kodak Germany site.) This evidently
means line widths. f50
expressed in the more familiar line pairs (or cycles)
per mm would be 55.

The likely reason that slide film has poorer sharpness is that
it's
made to be displayed directly. Dark tones must be deep and rich; Dmax
is
much higher than for negative films, hence more dye is needed. Emultion
layers may have to be thicker to accomodate this.
.

.

Why then do slide films have such a good reputation? Mostly
because
they have
finer grain (and also because
traditional printing loses some sharpness). Remember, sharpness
and grain are
not the same thing, although both
affect image quality. With
B&W films
you could trade them off with your choice of developer: Microdol-X had
the finest grain; Rodinal
was the sharpest but grainiest. Camera store/drugstore prints can't
begin
to do justice to the information stored on negatives. You need a scan,
perferably 4000 dpi.

As we mentioned
earlier, MTF (sharpness)
corresponds to the bandwidth of
a communications system while
grain corresponds to its
noise. Both enter into Shannon's
equation for information transmission capacity,

C
= ω log2(SNR+1)

where C is
capacity, ω is bandwidth,
and SNR is the signal-to-noise power ratio (drops
as grain increases).
I suspect that the tradeoff between grain and sharpness doesn't have
much
effect on capacity-- perceived image quality may be more a function of
C
than of either grain or sharpness by themselves. This, of course, is
mostly
speculation. I'll return to this topic in the discussion of image
quality
in Digital cameras vs. film

Film
links and MTF
data

Since
manufacturers keep altering their websites, I'll present a guide
to finding the data rather than the exact URL.

There is one troubling aspect to some manufacturer's
MTF curves, for
example Fujichrome
Provia 100F. MTF should be 100% at very low frequencies, but
it remains
flat at 120% from 10 lp/mm down to 1 lp/mm. There are three possible
reasons.
(1) The adjacency effect extends to extremely low spatial frequencies.
Doubtful. 1 lp/mm is awfully low. (2) The graphic arts department took
liberties with the plot before sending it out for publication. This
happened
to me in the old days before I had access to computer tools. (3) The
marketing
department cheated. It happens.

The Imatest
program allows you to measure the MTF of digital cameras or digitized
film images. You can't measure an individual component in isolation,
but you can compare components, such as lenses, with great accuracy. Imatest
also measurs other factors
that contribute to image quality.

Lenses

MTF
for film is rather simple-- it's the same in all directions and uniform
throughout the film surface. Not so lenses. MTF is a function of the
distance
from the image center, the aperture (f-stop), the spectrum of the
light, the focal length (for zooms), and even
the focusing distance (macro lenses are designed to maintain good MTF
at
close focus). Not only that, but there are two
MTF's at each point:
one along the radial (or sagittal) direction (pointing away from the
image
center) and one in the tangential direction (along a circle around the
image center), at right angles to the radial direction. Because of
this,
the MTF curve cannot be displayed in the same way as for film.

"The graphs show MTF in percent for the three line
frequencies of 10
lp/mm, 20 lp/mm and 40 lp/mm, from the center of the image (shown at
left)
all the way to the corner (shown at right). The top two lines represent
10 lp/mm, the middle two lines 20 lp/mm and the bottom two lines 40
lp/mm.
The solid lines represent sagittal MTF (lp/mm aligned like the spokes
in
a wheel). The broken lines represent tangential MTF (lp/mm arranged
like
the rim of a wheel, at right angles to sagittal lines). On the scale at
the bottom 0 represents the center of the image (on axis), 3 represents
3 mm from the center, and 21 represents 21 mm from the center, or the
very
corner of a 35 mm film image. Separate graphs show results at f8 and
full
aperture. For zoom lenses, there are graphs for each measured focal
length."

They state elsewhere that performance at 10 line
pairs/mm is indicative of
the lens contrast while 40 line pairs/mm is indicative of its sharpness.

Lens performance is typically limited by aberrations
at large apertures and diffraction
at small apertures. Aberrations depend on lens design and manufacturing
quality; they differ markedly for different lenses. Diffraction is a
fundamental physical effect; it depends on the aperture alone. A lens
is sharpest between the two extremes, near its optimim
aperture,
which tends to be around f/8 or f/11 for the 35mm format. It is smaller
(as low as f/4) for compact digital cameras and larger (f/11 to f/22)
for large format cameras.

Unlike film, the MTF of lenses don't necessarily
match the second order
1/(1+( f /f50)2)
equation. MTF rolloff can vary widely, depending on lens design and
manufacturing
tolerance. To accommodate this, MTFcurve has an input variable for the
order of the lens's MTF rolloff, lord, which
defaults to 2 if not
entered or if entered as 0. The lens MTF equation becomes,

MTFlens(f)
= 1/(1+|f/flens|lord)
(lord defaults to 2 in MTFcurve if not entered)

Flens is the frequency where
lens MTF = 0.5 = 50%, corresponding to f50
in film. We use lord = 2, which appears to be an
adequate approximation
for typical lenses, until better information is available. It's not
easy
to derive lord from Photodo's data because most of
the curves are
above MTF = 50%; you need to look at MTF below 30% to get a clear
picture
of the rolloff. Flens can be estimated by interpolating between curves
if MTF at 40 line pairs/mm (MTF40) is below 50%.
Or it can be estimated
from MTF40 using the following table, which is
based on the
inverse of the second order equation, flens = 40/sqrt(1/MTF40-1),
where MTF40 is expressed as a fraction rather
than a percentage.
Sqrt
can be replaced by exponentiation to the 1/lord
power ( flens =
40/(1/MTF40-1)^(1/lord) ) when lord
is not 2.

The
Canon
28-70mm f/2.8L lens in this test is an outstanding performer,
achieving
a Photodo
rating
of 3.9 out of a possible 5, about as good as zoom lenses for the 35mm
format (covering 43mm diagonal) get. At 40mm,
f/8, it's as sharp as a first rate prime (single focal length) lens.
With
MTF40 around 70% at f/8, flens is around 61 line
pairs/mm The down
side: it's expensive (over $1000 US), large, and heavy. But sharp. The
photo.net
review is full of hypebole like, "one *bleeping* sharp
zoom!," "sharpness
and contrast is spectacular," "a stunning piece of glass," etc. Since
35mm lenses don't get much better I use this lens as the standard for
the
"excellent 35mm lens" for the remainder of this article.

The plot on the right, generated by

MTFcurve2 45 13 61

shows the combined response of Velvia film and the
excellent 35mm lens.
The red
curve is the spatial response,
the blue
curve is the combined MTF,
and the thin blue
dashed curve is the
MTF of the lens only.

The 50% and 10% points for MTF are now 36.8 and 68.6
line pairs/mm. This
is the best that can be expected for an excellent
lens covering 43mm diagonal, optimum
aperture, correct focus, sturdy camera support, and good atmospheric
conditions.

Schneider Optics
has
a page on Quality
criteria of lenses that includes an explanation of MTF. If
you read
this page carefully, you'll find a minor error in their MTF diagram.
They
also publish MTF data for their large
format and enlarging
lenses. The curves look like Photodo's except that they're done at 5,
10,
and 20 line pairs per mm instead of 10, 20, and 40. I suppose this is
more appropriate
for large format. Unfortunately these curves are based on computer
simulation
rather than real measurements, which will always be a little worse.
Their
arch competitor Zeiss
has unkind things to
say about this technique.

Leica
has published
MTF data for several of its outstanding M-System
(rangefinder) and R-System
(SLR) lenses as PDF Product information, linked on the lower right of
these
two pages. MTF data is for 5, 10, 20, and 40 lp/mm. They state that the
5 and 10 lp/mm data relates to large object contrast while the 20 and
40
lp/mm data relates to small object resolution. It's instructive to
compare
Leica's data with detailed lens performance observations by Erwin
Puts.

Diffraction

Lenses
are sharpest between about two stops down from maximum aperture
and the aperture where diffraction, an unavoidable consequence of
physics,
starts to dominate. For 35mm lenses, this is typically between f/5.6
(f/8
for slow zooms) and f/11. At large apertures, resolution is limited by
aberrations (astigmatism, coma, etc.), which lens designers work
valiantly
to overcome. MTF wide open is almost always
poorer than MTF at f/8.

Diffraction worsens as the lens is stopped down (the f-stop is
increased).
The equation for the Rayleigh diffraction limit, adapted from R.
N. Clark's scanner detail page, is,

Rayleigh limit
(line pairs per mm) = 1/(1.22 Nω)

N is the f-stop setting and ω
=
the wavelength of light in
mm = 0.0005 mm for a typical daylight spectrum. (0.00055 mm is the
wavelength
of green light, where the eye is most sensitive, but 0.0005 mm may be
more
representative of daylight situations.) I've seen a simple rule of
thumb,
Rayleigh limit = 1600/N, which corresponds
to ω
= 0.000512
mm. The light circle formed by diffraction, known as the
Airy disk,
has a radius equal to1/(Rayleigh limit).

The MTF at the Rayleigh limit is about 9%. Significant
Rayleigh limits
are 149 lp/mm @ f/11, 102 lp/mm @ f/16, 74 lp/mm @ f/22, and 51 lp/mm @
f/32. Larry,
an experienced lens
designer, finds these numbers to be somewhat conservative because the
Rayleigh
limit is based on a spot, which has lower resolution than a band. His
numbers
of 125 lp/mm @ f/16 and 64 lp/mm @ f/32 are derived from a Kodak chart
he contributed to Robert
Monaghan's Lens Resolution Testing page. Most lenses are
aberration-limited
(relatively unaffected by diffraction) at f/8 and below. The OTF
(optical
transfer function) curve in David
Jacobson's Lens Tutorial shows how MTF (the magnitude of OTF)
varies
with spatial frequency for a purely diffraction-limited lens at f/22.

We can derive some interesting relationships from
David Jacobson's graph.
At the Rayleigh diffraction limit of 68 lp/mm (for f/22, ω
= 555
nm = 0.000555 mm), MTF is approximately 9%. It is 10% at about 64 lp/mm
and 50% at 32 lp/mm. The following relationships therefore hold for
diffraction-limited
lenses:

The diffraction curve is somewhat difficult to express mathematically,
but it can be approximated-- matched at the 10% and 50% MTF points-- by
the equation used in the MTFcurve program (f50
is the same as flens; f50
and lord are the third and fourth input arguments
to MTFcurve),

The question remains, at what f-stop does a lens become diffraction
limited?
The best estimate is the f-stop where the diffraction-limited f50
(0.38/(N ω)) equals flens--
the
50% MTF frequency at the
sharpest aperture. For the excellent 35mm lens, flens
= 61 lp/mm. The f-stop with the same diffraction-limited f50
is N = 0.38/(61*0.0005) = 12.5. We can therefore
say with some confidence
that good 35mm lenses are relatively unaffected by diffraction at f/8
and
below, moderately affected at f/11, and diffraction-limited at f/16 and
beyond. Such lenses should only be stopped down beyond f/11 (larger
f-stops
for larger formats) when extreme depth of field is required.
Diffraction
in digital cameras is discussed here.

The Imatest
program allows you to measure the MTF of digital cameras or digitized
film images. You can't measure an individual component in isolation,
but you can compare components, such as lenses, with great accuracy. Imatest
also measurs other factors
that contribute to image quality.

Large
format

Thanks
to Schneider's large
format lens data, we can compare large format image quality
to 35mm.
The MTF curve on the left is for the 150mm f/5.6 Schneider Apo-Symmar
lens,
focused at infinity at f/22. It could be a little better than actual
lens
performance because it's derived from computer simulation. But it's
almost
certainly worse than optimum because the lens is diffraction-limited at
f/22; it is almost certainly sharper at f/11 and f/16. Its image circle
is u' = 110mm, more than sufficient to cover the 4x5
(inch) format with room for camera movements. The principal difference
between this curve and the Photodo curves (above) is that the three
lines
(top to bottom) represent 5, 10, and 20 lines per mm (instead of 10,
20,
and 40). This is a first class view camera lens; hence we'll refer to
it
as an "excellent 4x5
lens."

We can find the lens's 50% MTF value, flens, by modifying the
above
equation to flens = 20/sqrt(1/MTF20-1), or we
can also use the
above table by substituting MTF20 for MTF40
and dividing
flens by 2. Since MTF20 ~= 66%, we estimate
flens to be 28 line pairs
per mm. This is just under half the value for the excellent 35mm lens
and
just slightly under the diffraction limit (f50
= 32 lp/mm for f/22). Even at optimum aperture (around f/11) a view
camera
lens is not likely to be as sharp as the excellent 35mm lens; it has to
cover about 4 times the image circle (16 times the area). Next we run MTFcurve
45 13 28, (Velvia film + lens) and we find that the 50%
and 10%
MTF values of the film + lens are 27.1 and 49.9 line pairs/mm.
Sharpness is
almost entirely limited by the lens; the film hardly plays a role.

Assuming a 35mm frame is cropped for an 8x10 print,
a 4x5
frame is 4 times larger. The ratio of total detail at the 50% level is
4*27.1/36.8 = 2.94. The ratio at the 10% level is 4*49.9/68.6 = 2.91. A
4x5 image can
therefore resolve approximately three
timesthe linear detail of 35mm, assuming both
employ good technique:
excellent lenses and film, optimum aperture, correct focus, sturdy
camera
support, good atmospheric conditions, etc. Since the passing of the
Speed
Graphic era, such good technique has been standard practice in large
format
photography; it's less common with 35mm. A 24x30 inch print from 4x5
would
have the same detail as an 8x10
from
35mm. It can be extremely sharp! This result is in
substantial agreement
with R.
N.
Clark's scanner detail page. If we are to believe Kodak's
T-MAX 100
data, the ratio would be lower: 35mm images would be phenomenally
sharp,
limited only by the lens.